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In this paper, we consider numerical approximations for solving the anisotropic Cahn–Hilliard model. We combine the Scalar Auxiliary Variable (SAV) approach with the stabilization technique to arrive at a novel Stabilized-SAV approach, where three linear stabilization terms, which are shown to be crucial to remove the oscillations caused by the anisotropic coefficient, are added to enhance the stability while keeping the required accuracy. The schemes are very easy-to-implement and fast in the sense that all nonlinear terms are treated in a semi-explicit way, and one only needs to solve three decoupled linear equations with constant coefficients at each time step. We further prove the unconditional energy stabilities rigorously and present numerous 2D and 3D numerical simulations to demonstrate the stability and accuracy.
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This content will become publicly available on January 30, 2026
Scalar Auxiliary Variable (SAV) Stabilization of Implicit‐Explicit (IMEX) Time Integration Schemes for Non‐Linear Structural Dynamics
Implicit-explicit (IMEX) time integration schemes are well suited for non-linear structural dynamics because of their low computational cost and high accuracy. However, the stability of IMEX schemes cannot be guaranteed for general non-linear problems. In this article, we present a scalar auxiliary variable (SAV) stabilization of high-order IMEX time integration schemes that leads to unconditional stability. The proposed IMEX-BDFk-SAV schemes treat linear terms implicitly using kth-order backward difference formulas (BDFk) and non-linear terms explicitly. This eliminates the need for iterations in non-linear problems and leads to low computational costs. Truncation error analysis of the proposed IMEX-BDFk-SAV schemes confirms that up to kth-order accuracy can be achieved and this is verified through a series of convergence tests. Unlike existing SAV schemes for first-order ordinary differential equations (ODEs), we introduce a novel SAV for the proposed schemes that allows direct solution of the second-order ODEs without transforming them into a system of first-order ODEs. Finally, we demonstrate the performance of the proposed schemes by solving several non-linear problems in structural dynamics and show that the proposed schemes can achieve high accuracy at a low computational cost while maintaining unconditional stability.
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- Award ID(s):
- 2229136
- PAR ID:
- 10574499
- Publisher / Repository:
- Wiley
- Date Published:
- Journal Name:
- International Journal for Numerical Methods in Engineering
- Volume:
- 126
- Issue:
- 2
- ISSN:
- 0029-5981
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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